Theorem etransclem27 42539

Description: The 𝑁-th derivative of 𝐹 applied to 𝐽 is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypotheses

Ref	Expression
etransclem27.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
etransclem27.x	⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
etransclem27.p	⊢ (𝜑 → 𝑃 ∈ ℕ)
etransclem27.h	⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
etransclem27.cfi	⊢ (𝜑 → 𝐶 ∈ Fin)
etransclem27.cf	⊢ (𝜑 → 𝐶:dom 𝐶⟶(ℕ0 ↑m (0...𝑀)))
etransclem27.g	⊢ 𝐺 = (𝑥 ∈ 𝑋 ↦ Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝑥))
etransclem27.jx	⊢ (𝜑 → 𝐽 ∈ 𝑋)
etransclem27.jz	⊢ (𝜑 → 𝐽 ∈ ℤ)

Assertion

Ref	Expression
etransclem27	⊢ (𝜑 → (𝐺‘𝐽) ∈ ℤ)

Distinct variable groups:   𝐶,𝑗,𝑙,𝑥   𝑥,𝐻   𝑗,𝐽,𝑙,𝑥   𝑗,𝑀,𝑥   𝑃,𝑗,𝑥   𝑥,𝑆   𝑗,𝑋,𝑥   𝜑,𝑗,𝑙,𝑥

Allowed substitution hints:   𝑃(𝑙)   𝑆(𝑗,𝑙)   𝐺(𝑥,𝑗,𝑙)   𝐻(𝑗,𝑙)   𝑀(𝑙)   𝑋(𝑙)

Proof of Theorem etransclem27

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		etransclem27.g	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝑋 ↦ Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝑥))
2		fveq2 6665	. . . . 5 ⊢ (𝑥 = 𝐽 → (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝑥) = (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽))
3	2	prodeq2ad 41865	. . . 4 ⊢ (𝑥 = 𝐽 → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝑥) = ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽))
4	3	sumeq2sdv 15055	. . 3 ⊢ (𝑥 = 𝐽 → Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝑥) = Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽))
5		etransclem27.jx	. . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑋)
6		etransclem27.cfi	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Fin)
7		dmfi 8796	. . . . 5 ⊢ (𝐶 ∈ Fin → dom 𝐶 ∈ Fin)
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → dom 𝐶 ∈ Fin)
9		fzfid 13335	. . . . 5 ⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → (0...𝑀) ∈ Fin)
10		etransclem27.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
11	10	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
12		etransclem27.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
13	12	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
14		etransclem27.p	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℕ)
15	14	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
16		etransclem27.h	. . . . . . . 8 ⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
17		etransclem5 42517	. . . . . . . 8 ⊢ (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑧 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑧)↑if(𝑧 = 0, (𝑃 − 1), 𝑃))))
18	16, 17	eqtri 2844	. . . . . . 7 ⊢ 𝐻 = (𝑧 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑧)↑if(𝑧 = 0, (𝑃 − 1), 𝑃))))
19		simpr 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
20		etransclem27.cf	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶:dom 𝐶⟶(ℕ0 ↑m (0...𝑀)))
21	20	ffvelrnda 6846	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → (𝐶‘𝑙) ∈ (ℕ0 ↑m (0...𝑀)))
22		elmapi 8422	. . . . . . . . 9 ⊢ ((𝐶‘𝑙) ∈ (ℕ0 ↑m (0...𝑀)) → (𝐶‘𝑙):(0...𝑀)⟶ℕ0)
23	21, 22	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → (𝐶‘𝑙):(0...𝑀)⟶ℕ0)
24	23	ffvelrnda 6846	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶‘𝑙)‘𝑗) ∈ ℕ0)
25	11, 13, 15, 18, 19, 24	etransclem20 42532	. . . . . 6 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗)):𝑋⟶ℂ)
26	5	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝐽 ∈ 𝑋)
27	25, 26	ffvelrnd 6847	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℂ)
28	9, 27	fprodcl 15300	. . . 4 ⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℂ)
29	8, 28	fsumcl 15084	. . 3 ⊢ (𝜑 → Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℂ)
30	1, 4, 5, 29	fvmptd3 6786	. 2 ⊢ (𝜑 → (𝐺‘𝐽) = Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽))
31	11, 13, 15, 18, 19, 24, 26	etransclem21 42533	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) = if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))))))
32		iftrue 4473	. . . . . . . 8 ⊢ (if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))))) = 0)
33		0zd 11987	. . . . . . . 8 ⊢ (if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗) → 0 ∈ ℤ)
34	32, 33	eqeltrd 2913	. . . . . . 7 ⊢ (if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))))) ∈ ℤ)
35	34	adantl 484	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))))) ∈ ℤ)
36		0zd 11987	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → 0 ∈ ℤ)
37		nnm1nn0 11932	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ0)
38	14, 37	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ0)
39	14	nnnn0d 11949	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ ℕ0)
40	38, 39	ifcld 4512	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
41	40	nn0zd 12079	. . . . . . . . . . . . . 14 ⊢ (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ)
42	41	ad3antrrr 728	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ)
43	24	nn0zd 12079	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶‘𝑙)‘𝑗) ∈ ℤ)
44	43	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((𝐶‘𝑙)‘𝑗) ∈ ℤ)
45	42, 44	zsubcld 12086	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ ℤ)
46	36, 42, 45	3jca 1124	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (0 ∈ ℤ ∧ if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ ℤ))
47	44	zred 12081	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((𝐶‘𝑙)‘𝑗) ∈ ℝ)
48	42	zred 12081	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
49		simpr 487	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗))
50	47, 48, 49	nltled 10784	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((𝐶‘𝑙)‘𝑗) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))
51	48, 47	subge0d 11224	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ↔ ((𝐶‘𝑙)‘𝑗) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃)))
52	50, 51	mpbird 259	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → 0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))
53		0red 10638	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 0 ∈ ℝ)
54	24	nn0red 11950	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶‘𝑙)‘𝑗) ∈ ℝ)
55	40	nn0red 11950	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
56	55	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
57	24	nn0ge0d 11952	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 0 ≤ ((𝐶‘𝑙)‘𝑗))
58	53, 54, 56, 57	lesub2dd 11251	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − 0))
59	56	recnd 10663	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℂ)
60	59	subid1d 10980	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − 0) = if(𝑗 = 0, (𝑃 − 1), 𝑃))
61	58, 60	breqtrd 5085	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))
62	61	adantr 483	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))
63	46, 52, 62	jca32 518	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((0 ∈ ℤ ∧ if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ ℤ) ∧ (0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))))
64		elfz2 12893	. . . . . . . . . . 11 ⊢ ((if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ (0...if(𝑗 = 0, (𝑃 − 1), 𝑃)) ↔ ((0 ∈ ℤ ∧ if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ ℤ) ∧ (0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))))
65	63, 64	sylibr 236	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ (0...if(𝑗 = 0, (𝑃 − 1), 𝑃)))
66		permnn 13680	. . . . . . . . . 10 ⊢ ((if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ (0...if(𝑗 = 0, (𝑃 − 1), 𝑃)) → ((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) ∈ ℕ)
67	65, 66	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) ∈ ℕ)
68	67	nnzd 12080	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) ∈ ℤ)
69		etransclem27.jz	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ ℤ)
70	69	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → 𝐽 ∈ ℤ)
71		elfzelz 12902	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
72	71	ad2antlr 725	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → 𝑗 ∈ ℤ)
73	70, 72	zsubcld 12086	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (𝐽 − 𝑗) ∈ ℤ)
74		elnn0z 11988	. . . . . . . . . 10 ⊢ ((if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ ℕ0 ↔ ((if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ ℤ ∧ 0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))))
75	45, 52, 74	sylanbrc 585	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ ℕ0)
76		zexpcl 13438	. . . . . . . . 9 ⊢ (((𝐽 − 𝑗) ∈ ℤ ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ ℕ0) → ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))) ∈ ℤ)
77	73, 75, 76	syl2anc 586	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))) ∈ ℤ)
78	68, 77	zmulcld 12087	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) ∈ ℤ)
79	36, 78	ifcld 4512	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))))) ∈ ℤ)
80	35, 79	pm2.61dan 811	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))))) ∈ ℤ)
81	31, 80	eqeltrd 2913	. . . 4 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℤ)
82	9, 81	fprodzcl 15302	. . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℤ)
83	8, 82	fsumzcl 15086	. 2 ⊢ (𝜑 → Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℤ)
84	30, 83	eqeltrd 2913	1 ⊢ (𝜑 → (𝐺‘𝐽) ∈ ℤ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ifcif 4467  {cpr 4563   class class class wbr 5059   ↦ cmpt 5139  dom cdm 5550  ⟶wf 6346  ‘cfv 6350  (class class class)co 7150   ↑_m cmap 8400  Fincfn 8503  ℂcc 10529  ℝcr 10530  0cc0 10531  1c1 10532   · cmul 10536   < clt 10669   ≤ cle 10670   − cmin 10864   / cdiv 11291  ℕcn 11632  ℕ₀cn0 11891  ℤcz 11975  ...cfz 12886  ↑cexp 13423  !cfa 13627  Σcsu 15036  ∏cprod 15253   ↾_t crest 16688  TopOpenctopn 16689  ℂ_fldccnfld 20539   D^𝑛 cdvn 24456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-inf2 9098  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609  ax-addf 10610  ax-mulf 10611

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-iin 4915  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-se 5510  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-isom 6359  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-om 7575  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-pm 8403  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fsupp 8828  df-fi 8869  df-sup 8900  df-inf 8901  df-oi 8968  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-q 12343  df-rp 12384  df-xneg 12501  df-xadd 12502  df-xmul 12503  df-icc 12739  df-fz 12887  df-fzo 13028  df-seq 13364  df-exp 13424  df-fac 13628  df-bc 13657  df-hash 13685  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-clim 14839  df-sum 15037  df-prod 15254  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-plusg 16572  df-mulr 16573  df-starv 16574  df-sca 16575  df-vsca 16576  df-ip 16577  df-tset 16578  df-ple 16579  df-ds 16581  df-unif 16582  df-hom 16583  df-cco 16584  df-rest 16690  df-topn 16691  df-0g 16709  df-gsum 16710  df-topgen 16711  df-pt 16712  df-prds 16715  df-xrs 16769  df-qtop 16774  df-imas 16775  df-xps 16777  df-mre 16851  df-mrc 16852  df-acs 16854  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-submnd 17951  df-mulg 18219  df-cntz 18441  df-cmn 18902  df-psmet 20531  df-xmet 20532  df-met 20533  df-bl 20534  df-mopn 20535  df-fbas 20536  df-fg 20537  df-cnfld 20540  df-top 21496  df-topon 21513  df-topsp 21535  df-bases 21548  df-cld 21621  df-ntr 21622  df-cls 21623  df-nei 21700  df-lp 21738  df-perf 21739  df-cn 21829  df-cnp 21830  df-haus 21917  df-tx 22164  df-hmeo 22357  df-fil 22448  df-fm 22540  df-flim 22541  df-flf 22542  df-xms 22924  df-ms 22925  df-tms 22926  df-cncf 23480  df-limc 24458  df-dv 24459  df-dvn 24460

This theorem is referenced by: (None)